Sur Spinoza

Cours Vincennes : the actual infinite-eternal, the logic of relations
Cours du 10/03/1981
Simon Duffy,

Confrontation with Gueroult's commentary.

This week and next week I will be speaking again of Spinoza, and then that‚s it. Unless you have questions to pose, which I would like very much.
Ok then: my dream is that it is very clear for you, this conception of individuality, such that we‚ve tried to bring it out in the philosophy of Spinoza, because, finally, it seems to me that this is one of the newest elements of Spinozism. It is this manner in which the individual, as such, is going to be conveyed, related, reported in Being. And in order to try to make you understand this conception of individuality which seems to me so new in Spinoza, I will always return to the theme: it is as if an individual, whatever individual, had three layers, as if it was composed, then, of three layers. We have advanced, at least into the first dimension, into the first layer of the individual, and I say: oh yes, all individuals have an infinity of extensive parts. This is the first point: an infinity of extensive parts. In other words, there are only individuals that are composite. A simple individual, I believe that, for Spinoza, it is a notion lacking in sense.
Every individual, as such, is composed of an infinity of parts.
I‚ll try to summarize very quickly: What does this mean this idea that the individual is composed of an infinity of parts? What are these parts? Once again, they are what Spinoza calls Œthe simplest bodies‚: all bodies are composed of an infinity of very simple bodies. But what are these: Œvery simple bodies‚? We have arrived at a precise enough status: they are not atoms, meaning finite bodies, and neither are they indefinites. What are they? And there Spinoza belongs to the 17th century. Once again, what really strikes me, in regard to the thought of the 17th century, is the impossibility of grasping this thought if we don‚t take into account one of the richest notions of this era, a notion which is simultaneously metaphysical, physical, mathematical, etc: the notion of the Œactual infinite‚. Now the actual infinite is neither finite nor indefinite. The finite signifies, above all, it refers to, if I seek the formula of the finite, it is: there is a moment where you have to stop yourself. That is to say: when you analyse something there will always be a moment where it will be necessary to stop yourself. Let‚s say, and for a long time, this moment of the finite, this fundamental moment of the finite which marks the necessity of finite terms, it is all of this which inspired atomism since Epicurus, since Lucretius: the analysis encounters a limit, this limit is the atom. The atom is subject to a finite analysis. The indefinite is as far as you can go, you can‚t stop yourself. That is to say: as far as you can take the analysis, the term at which you arrive will always be, in turn, divided and analysed. There will never be a last term.
The point of view of the actual infinite, it seems to me, of which we have completely lost the sense, and we have lost this sense for a thousand reasons, I suppose, among others for scientific reasons, all this ? But what matters to me, is not why we have lost this sense, it is as if I have happened to be able to reconstruct for you the way in which these thinkers thought. Really, it is fundamental in their thinking. Once again, if I consider that Pascal wrote texts that are representative of the 17th Century, these are essentially texts on man in relation to the infinite. These are people who truly thought naturally, philosophically, in terms of the actual infinite. Now this idea of an actual infinite, that is to say neither finite nor indefinite, what does that tell us? What it tells us is that: there are last terms, there are ultimate terms ˜ you see, this is contrary to the indefinite, it is not the indefinite since there are ultimate terms, only these ultimate terms are ad infinitum. Therefore, they are not the atom. They are neither finite nor indefinite. The infinite is actual, the infinite is in action. In effect, the indefinite is, if you like, infinite, but virtual, that is to say: you can always go further. This is not it; it (the actual infinite) tells us: there are last terms: Œthe simplest bodies‚ for Spinoza. These are the ultimate terms, these are the terms which are last, which you can no longer divide. But, these terms are infinitely small. They are the infinitely small, and this is the actual infinite. Note that it is a struggle on two fronts: simultaneously against finitism and against the indefinite. What does this mean? There are ultimate terms, but these are not atoms since they are the infinitely small, or as Newton will say, they are vanishings, vanishing terms. In other words, smaller than any given quantity. What does this imply? Infinitely small terms; you can‚t treat them one by one. This too is a non-sense: to speak of an infinitely small term that I would consider singularly, that makes no sense. The infinitely small, they can only go by way of infinite collections. Therefore there are infinite collections of the infinitely small. The simple bodies of Spinoza don‚t exist one by one. They exist collectively and not distributively. They exist by way of infinite sets. And I cannot speak of a simple body, I can only speak of an infinite set of simple bodies. Such that an individual is not a simple body, an individual, whatever it is, and however small it is, an individual has an infinity of simple bodies, an individual has an infinite collection of the infinitely small.
That is why, despite all the force of Gueroult‚s commentary on Spinoza, I cannot understand how Gueroult poses the question of knowing if simple bodies for Spinoza shouldn‚t have shape and magnitude ? It is obvious that if simple bodies are infinitely small, that is to say, "vanishing‰ quantities, they have neither shape nor magnitude. An atom, yes, has a shape and a magnitude: it is smaller than any given magnitude. What then has shape and magnitude? What has shape and magnitude, here, the response is very simple, what has shape and magnitude is a collection, it is a collection itself infinite of the infinitely small. This yes, the infinite collection of the infintely small has shape and magnitude. However, we come up against this problem: yes, but where does it come from, this shape and this magnitude? I mean: if the simple bodies are all infinitely small, what permits us to distinguish such an infinite collection of the infinitely small from another such infinite collection of the infinitely small? From the point of view of the actual infinite, how can we make distinctions in the actual infinite? Or even: Is there only one collection? One collection of all the possible infinitely smalls? Now Spinoza is very firm here! He says: to each individual corresponds an infinite collection of very simple bodies, each individual is composed of an infinity of very simple bodies. It is necessary therefore that I have the means of recognising the collection of the infinitely small that corresponds to such an individual, and that which corresponds to another such individual. How is that to be done? Before arriving at this question, let‚s try to see how these infinitely small are. They enter therefore into infinite collections, and I believe that here the 17th century grasped something that mathematics, with completely different means, with a completely different procedure˜ I don‚t want to make arbitrary connections ˜ but that modern mathematics rediscovered with a completely different procedure, that is to say: a theory of infinite sets. The infinitely small enter into infinite sets and these infinite sets are not the same. That is to say: there is a distinction between infinite sets. Regardless of whether it was Leibniz, or Spinoza, the second half of the 17th century is riddled with this idea of the actual infinite, the actual infinite which consists of these infinite sets of the infinitely small.
But then these vanishing terms, these infinitely small terms, what are their ? How are they? I would like this to take on a slightly more concrete shape. It is obvious that they don‚t have interiority. I‚ll try first to say what they are not, before saying what they are. They have no interiority, they enter into infinite sets, the infinite set could have an interiority. But these extreme terms, infinitely small, vanishing, they have no interiority, they are going to constitute what? They are going to constitute a veritable matter of exteriority. The simple bodies have only strictly extrinsic relations, relations of exteriority with each other. They form a species of matter, using Spinoza‚s terminology: a modal matter, a modal matter of pure exteriority, which is to say: they react on one another, they have no interiority, they have only external relations with one another. However, I‚ll return to my question: if they have only external relations, what allows us to distinguish one infinite set from another? Once again, all individuals, each individual, here I can say each individual since the individual isn‚t the very simple body, each individual, distributively, has an infinite set of infinitely small parts. These parts are actually given. But what distinguishes my infinite set, the infinite set that refers to me, and the set that refers to a neighbour? Hence, and already we are entering the second layer of individuality, which leads us to ask: under what aspect does an infinite set of very simple bodies belong to either this or that individual? Under what aspect?
It is understood, I have an infinite eset of infinitely small parts, but under what aspect does that infinite set belong to me? Under what aspect does an infinite set of very simple bodies belong to either this or that individual. It is understood that I have an infinite set of infinitely small parts, but under what aspect does that infinite set belong to me?
You see that I have only with difficulty transformed the question because when I ask under what aspect the infinite set belongs to me, it is another way of asking what allows me to distinguish such an infinite set from another such infinite set. Once again, at first sight, in the infinite everything must be confused, it must be the black night or the white light. What makes it so that I can distinguish infinities from one another? Under what aspect is an infinite set said to belong to me or to someone else?
Spinoza‚s response seems to be: an infinite set of infinitely small parts belongs to me, and not to someone else, insofar as this infinite set puts into effect [effectue] a certain relation. It is always under a relation that the parts belong to me. To the point that, if the parts which compose me take on another relation, at that very moment, they no longer belong to me. They belong to another individuality, they belong to another body. Hence the question: what is this relation? Under what relation can the infinitely small elements be said to belong to something? If I answer the question, I truly have the answer that I‚m looking for. I will show how, according to which condition, an infinite set can be said to belong to a finite individuality. Under what relation of the infinitely small can they belong to a finite individuality? Good. Spinoza‚s answer, if I stick to the letter of Spinoza, is: under a certain relation of movement and rest. But we‚re already there, relation of movement and rest, we know that it doesn‚t at all mean ˜ and here we would be wrong to read the text too quickly ˜ it doesn‚t at all mean, as in Descartes, a sum (which we have seen: the relation of movement and rest, this cannot be the Cartesian formula ? = mv, mass-velocity). No, he didn‚t say "relation‰. What defines the individual, is therefore a relation of movement and rest because it is under this relation that an infinity of infinitely small parts belong to the individual. However, what is this relation of movement and rest that Spinoza invokes in such a way?
Here, I recommence a confrontation with Gueroult‚s commentary. Gueroult makes an extremely interesting hypothesis, but here too I don‚t understand; I don‚t understand why he makes this hypothesis here, but it is very interesting. He says: finally the relations of movement and rest is vibration. At the same time it is a response that appears to me to be very curious. The answer must be very precise: it is a vibration! What does that mean? That would mean that what defines the individual, at the level of the second layer, that is to say the relation under which the parts belong to it, is a way of vibrating. Each individual ? Well, that would be good, that would be very concrete, what would define you, me, is that we would have a kind espèce of way of vibrating. Why not? Why not ? what does that mean? Either it is a metaphor, or else it means something. A vibration returns us to what, physics? It returns to something simpler, to a well known phenomenon which is that of the pendulum. Well, Gueroult‚s hypothesis seems to take on a sense that‚s very interesting because physics, in the 17th century, had considerably advanced the study of rotating bodies and pendulums, and notably had established a distinction between simple pendulums and compound pendulums. Well then good ? at this moment then you see that the Gueroult hypothesis becomes this: each simple body is a simple pendulum, and the individual, which has an infinity of simple bodies, is a compound pendulum. We would all be compound pendulums. That‚s very good! Or turning discs. It is an interesting conception of each of us. What does it mean? In effect, a simple pendulum is defined by what? It is defined, if you vaguely recall memories of physics, but very simple physics, it is defined in a certain way by a time, a time of vibration or a time of oscillation. For those who remember, there is the famous formula: t = py root of 1 over g [ ]? Yes, I think so. "t‰ is the duration of oscillation, "l‰ is the length of the thread on which the pendulum is suspended, "g‰ is what, in the 17th century, is called the intensity of gravity, it‚s of little significance ? Good. What is important is that in the formula, see that a simple pendulum has a time of oscillation which is independent of the amplitude of oscillation of the shaft of the pendulum, therefore completely independent of the amplitude of oscillation, independent of the mass of the pendulum ˜ this responds well to the situation of an infinitely small body, and independent of the weight of the thread. Weight of the thread, mass of the pendulum, only enter into play from the point of view of the compound pendulum. Therefore it seems that, in many respects, the Gueroult hypothesis works. Individuals for Spinoza would be kinds of compound pendulums, each composed of an infinity of simple pendulums. And what defines an individual is a vibration. Good.
Well then I say with a lot of freedom, like that, I am developing this for those who are very technically interested in Spinoza, as for the others you can retain what you want ? At the same time it is curious because, at the same time this hypothesis draws my attention, and I can‚t well see why. There is one thing which disturbs me: it is that it is true that all of the history of pendulums and of rotating discs, in the 17th century, is very encouraging; but precisely, if it had been this that Spinoza had wanted to say, why did he make no allusion to these problems of vibration, even in his letters? And then, above all, the model of the pendulum does not give a full account of what appears to me the essential, that is to say: this presence of the actual infinite and the term "infinitely small‰.
You see Gueroult‚s answer, insofar as he comments on Spinoza: the relation of movement and rest must be understood as the vibration of the simple pendulum. There you are! I am not at all saying that I am right, truly not ? I mean: if it is true that the very simple bodies ˜ this is why elsewhere Gueroult needs to affirm that the very simple bodies have nevertheless, for Spinoza, a shape and a magnitude. Suppose on the contrary ˜ but I am not at all saying that I am right˜ suppose that the very simple bodies were really infinitely small, that is to say that they have neither shape nor magnitude. At that moment then the model of the simple pendulum cannot work, and it cannot be a vibration that defines the relation of movement and rest.
On the other hand, we have another way, and then you can perhaps find others ˜ surely you can find others. The other way would be this: once again I return to my question, between supposedly infinitely small terms, what type of relations can they have? The response is very simple: between infinitely small terms, if we understand what is meant in the 17th century by the infinitely small, that is: which have no distributive existence, but which necessarily enter into an infinite collection, between infinitely small terms, there can only be one type of relation: differential relations.
Why? Infinitely small terms are vanishing terms, that is to say, the only relations which they can have between the infinitely small terms are relations which subsist while the terms vanish. A very simple question is: what are relations such that they subsist while their terms vanish? Let‚s do here some very very simple mathematics. I view , if I remain there in the 17th century and in a certain state of mathematics, and what I am saying is very rudimentary, I view as well known in the 17th century three types of relation. There are fractional relations which have been known for a very very long time; there are algebraic relations which are known ˜ which were anticipated well before, that goes without saying ˜ but which received a very firm status, in the 16th and 17th century ˜ in the 17th century with Descartes, that is in the first half of the 17th century, with algebraic relations; and finally differential relations, which at the moment of Spinoza and Leibniz, are the big question of mathematics of this era. I‚ll give some examples: I want it to be clear for you, even if it is not mathematics that I‚m doing, not at all. Example of a fractional relation: 2/3. Example of an algebraic relation: ax+by = etc. From which you can get x/y =. Example of a differential relation, we have seen: dy/dx = z. Good. What difference is there between these three types of relation? I would say that the fractional relation is already very interesting because otherwise we could make like a scale: the fractional relation is irreducibly a relation. Why?
If I say 2/3, 2/3, once again it is not a number. Why is it that 2/3 is not a number, it is because there isn‚t a number assignable which multiplied by 3 gives 2. Therefore it is not a number. A fraction is not a number, it is a complex of numbers, which I decide, by convention, to treat as a number; that‚s to say that I decide by convention to submit to the rules of addition, of subtraction, of multiplication. But a fraction is obviously not a number. Once I have found the fraction, I can treat numbers like fractions, that‚s to say: once I employ fractional symbolism, I can treat a number, for example the number 2, as a fraction. I can always write 4 over 2. 4 over 2 = 2. But the fractions, in their irreducibility to whole numbers, don‚t have numbers, but are complexes of whole numbers. Good. Therefore already the fraction brings forward a sort of independence of the relation in relation to its terms.
In this very important question of a logic of relations, the point of departure of a logic of relations is obvious: in what sense is there a consistency of the relation independent of its terms? The fractional number already gives me a kind of first approximation, but that doesn‚t allow us to avoid the fact that in the fractional relation, the terms must again be specified. The terms must be specified, that‚s to say that you could always write 2 over 3, but the relation is between two terms: 2 and 3. It is irreducible to these terms since it is itself not a number but a complex of numbers; but the terms must be specified, the terms must be given. In a fraction, the relation is as independent of its terms, Yes! But the terms must be given.
One step further. When I take an algebraic relation of the type x over y, this time I don‚t have given terms, I have two variables. I have variables. You See that everything happens as if the relation had acquired a superior degree of independence in relation to its terms. I no longer need to assign a determinate value. In a fractional relation I cannot escape this: I must assign a determinate value to the terms of the relation. In an algebraic relation I no longer need to assign a determinate value to the terms of the relation. The terms of the relation are variable. But that doesn‚t allow me to avoid the fact that it is again necessary that my variables have a determinable value. In other words, x and y can have all sorts of singular values, but they must have one. See, in the fractional relation, I can only have a singular value, or equivalent singular values. In an algebraic relation I no longer have to have a singular value, but that doesn‚t allow me to avoid the fact that my terms continue to have a specifiable value, and the relation is quite independent of every particular value of the variable, but it is not independent of a determinable value of the variable.
What is very new with the differential relation is that it takes something like a third step. When I say dy over dx, remember what we saw: dy in relation to y = 0; it is an infinitely small quantity. Dx in relation to x equals zero; therefore I can write, and they wrote constantly in the 17th century, in this form: dy over dx = 0 over 0: dy/dx=0/0. Now, the relation 0 over 0 is not equal to zero. In other words, when the terms vanish, when the terms vanish, the relation subsists. This time, the terms between which the relation is established are neither determined, nor determinable. Only the relation between its terms is determined. It is here that logic is going to make a leap, but a fundamental leap. Under this form of the differential calculus is discovered a domain where the relations no longer depend on their terms: the terms are reduced to vanishing terms, to vanishing quantities, and the relation between these vanishing quantities is not equal to zero. To the point where I would write, here I‚ll do it very summarily: dy over dx equals z: dy/dx = z. What does this mean "= z‰? It means, of course, that the differential relation dy over dx [dy/dx], which is made between vanishing quantities of Œy‚ and vanishing quantities of Œx‚, tells us strictly nothing about Œx‚ and Œy‚, but tells us something about Œz‚. For example, as applied to a circle, the differential relation dy over dx tells us something about a tangent called the "trigonometric tangent‰. In order to keep it simple, there is no need to understand anything, I can therefore write dy/dx = z. What does this mean then? See that the relation such as it subsists when its terms vanish is going to refer to a third term, Œz‚. It is interesting; this must have been very interesting: it is from here that a logic of relations is possible. What does this mean then? We will say of Œz‚ that it is the limit of the differential relation. In other words, the differential relation tends towards a limit. When the terms of the relation vanish, Œx‚ and Œy‚, and become dy and dx, when the terms of the relation vanish, the relation subsists because it tends towards a limit, Œz‚. When the relation is established between infinitely small terms, it does not cancel itself out at the same time as its terms, but tends towards a limit. This is the basis of differential calculus such as it is understood or interpreted in the 17th century.
Now you obviously understand why this interpretation of the differential calculus is at one with the understanding of an actual infinite, meaning with the idea of infinitely small quantities of vanishing terms.
Now, me, my answer to the question: but what is it exactly, this that Spinoza speaks to us of when he speaks of the relations of movement and rest, of proportions of movement and rest, and says: the infinitely small, a collection of the infinitely small belonging to such an individual under such a relation of movement and rest, what is this relation? I would not be able to say like Gueroult that it is a vibration which assimilates the individual to a pendulum: it is a differential relation. It is a differential relation such that it is manifested in the infinite sets, in the infinite sets of the infinitely small. And, in effect, if you take Spinoza‚s letter on blood, of which I have made great use, and the two components of blood, chyle and lymph, this now tells us what? It tells us that there are corpuscles of chyle, or better chyle is an infinite set of very simple bodies. Lymph is another infinite set of the very simple bodies. What distinguishes the two infinite sets? It is the differential relation! You have this time a dy/dx which is: the infinitely small parts of chyle over the infinitely small parts of lymph, and this differential relation tends towards a limit: the blood, that is to say: chyle and lymph compose blood.
If this is right, we could say why infinite ensembles are distinguished. It is because the infinite sets of very simple bodies don‚t exist independently of the differential relations which they put into effect. Therefore it is by abstraction that I began by speaking of them. But they necessarily exist, they exist necessarily under such and such a variable relation, they cannot exist independently of a relation, since the notion even of the term infinitely small, or of vanishing quantity, cannot be defined independently of a differential relation. Once again, Œdx‚ has no sense in relation to Œx‚, Œdy‚ has no sense in relation to Œy‚, only the relation dx/dy has a sense. That‚s to say that the infinitely small don‚t exist independently of the differential relation. Good. Now, what permits me to distinguish one infinite set from another infinite set? I would say that the infinite sets have different powers [puissances], and that which appears quite obviously in this thought of the actual infinite is the idea of the power [puissance] of an set. Let‚s understand here that I don‚t at all mean, it would be abominable to make me mean that they have anticipated things which closely concern set theory in the mathematics of the beginning of the 20th century, I don‚t mean that at all. I mean that in their conception, which is in absolute contrast with modern mathematics, which is completely different, which has nothing to do with modern mathematics, in their conception of the infinitely small and of the differential calculus interpreted from the perspective of the infinitely small, they necessarily brought out ˜ and this is not peculiar to Leibniz, it is also true of Spinoza, and of Malebranche, all these philosophers of the second half of the 17th century ˜ brought out the idea of infinite sets which are distinguished, not by their numbers, an infinite set by definition, it can not be distinguished from another infinite set by the number of its parts, since all infinite sets excede all assignable number of parts ˜ therefore, from the point of view of the number of parts, there cannot be one which has a greater number of parts than another. All these sets are infinite. Therefore under what aspect are they distinguished? Why is it that I can say: this infinite set and not that one?
I can say it, it is quite simple: because infinite sets are defined as infinite under such and such a differential relation. Between other terms the differential relations can be considered as the power [puissance] of an infinite set. Because of this an infinite set will be able to be of a higher power [puissance] than another infinite set. It‚s not that it will have more parts, obviously not, but it is that the differential relation under which the infinity, the infinite set of parts, belongs to it will be of higher power [puissance] than the relation under which an infinite set belongs to another individual ? [end of tape]
If we eliminate that, any idea of an actual infinite makes no sense. It is for this reason that, with the reservations that I‚ve just mentioned, for my part, the answer that I would give to: what is this relation of movement and rest that is for Spinoza characteristic of the individual, that is as the second layer of the individual, I would say that, no, it is not exactly a way of vibrating, perhaps we could bring together the two points of view, I don‚t know, but it is differential relation, and it is the differential relation that defines power [puissance]. Now, you understand the situation, yes? ˜ you recall that the infinitely small are constantly influenced from the outside, they pass their time by being in relation with other collections of the infinitely small. Suppose that an infinite collection of the infinitely small is determined from the outside to take another relation than the one under which it belongs to me. What does this mean? It means that: I die! I die! In effect, the infinite set which belongs to me under such a relation which characterises me, under my characteristic relation, this infinite set will take another relation under the influence of external causes. Take again the example of poison which decomposes the blood: under the action of arsenic, the infinitely small particles which compose the blood, which compose my blood under such a relation, are going to be determined to enter under another relation. Because of this, this infinite set is going to enter in the composition of another body, it will no longer be mine: I die! Do you understand? Good. If all of this is true, if it is true ? we are still missing something, because this relation, it comes from where, this relation? You can see that I‚ve progressed, but it is necessary for me to have my three layers. I cannot pull through in any other way. I need my three layers because I can‚t otherwise pull through. I start by saying: I am composed of an infinity of vanishing and infinitely small parts. Good. But be careful, these parts belong to me, they compose me under a certain relation which characterises me. But this relation which characterises me, this differential relation or better, this summation, not an addition but this kind of integration of differential relations, since in fact there are an infinity of differential relations which compose me: my blood, my bones, my flesh, all this refers to all sorts of systems of differential relations. These differential relations that compose me, that is to say which determine that the infinite collections which compose me belong effectively to me, and not to another, insofar as it endures, since it always risks no longer enduring, if my parts are determined to enter under other relations, they desert my relation. Ha! ? they desert my relation. Once again: I die! But that is going to involve lots of things.
What does it mean to die, at that very moment? It means that I no longer have parts. It‚s stupefying! Good. But this relation which characterises me, and which determines that the parts which put into effect the relation belong to me as soon as they put into effect the relation, insofar as they put into effect the differential relation, they belong to me, this differential relation, is this the last word on the individual? Obviously not, it is necessary to give an account of it in its turn. What is it going to express, it depends on what? What does it do that ? it doesn‚t have its own reason, this differential relation. What does it do that, me, I am characterised by such a relation or such an set of relations?
Last layer of the individual, Spinoza‚s answer: it is that the characteristic relations which constitute me, that is to say which determine that the infinite sets which verify these relations, which put into effect these relations which belong to me, the characteristic relations express something. They express something which is my singular essence. Here Spinoza says this very firmly: the relations of movement and rest serve only to express a singular essence. That means that none of us have the same relations, of course, but it isn‚t the relation that has the last word. It‚s what? Couldn‚t we here return to something of Gueroult‚s hypothesis? Last question: there is therefore a last layer of the individual, that is to say that the individual is a singular essence. You can now see what formula I can give to the individual: each individual is a singular essence, each singular essence expresses itself in the characteristic relations of the differential relation type, and under these differential relations, the infinite collections of the infinitely small belong to the individual.
Hence a last question: what is it, this singular essence? Couldn‚t we find here, at this level ˜ though it would be necessary to say that Gueroult, in all rigour, is mistaken about this level ˜ at this level something equivalent to the idea of vibration? What is a singular essence? Be careful that you have understood the question, it is almost necessary to consent to press the conditions of such a question. I am no longer in the domain of existence. What is it, existence? What does it mean, to me, to exist? We will see that it is just as complicated in Spinoza, because he gives a very rigorous determination to what he calls existing. But if we start with the most simple, I would say: to exist is to have an infinity of extensive parts, of extrinsic parts, to have an infinity of infinitely small extrinsic parts, which belong to me under a certain relation. Insofar as I have, in effect, extensive parts which belong to me under a certain relation, infinitely small parts which belong to me, I can say: I exist.
When I die, once again, then it is necessary to work out the Spinozist concepts, when I die what happens? To die means, exactly this, it means: the parts which belong to me cease to belong to me. Why? We have seen that they only belong to me insofar as they put into effect a relation, relation which characterises me. I die when these parts which belong to me or which belonged to me are determined to enter under another relation which characterises another body: I would feed worms! "I would feed worms‰, which means: the parts of which I am composed enter under another relation ˜ I am eaten by worms. My corpuscles, mine, which pass under the relation of the worms. Good! That could happen ? Or better, the corpuscles of which I am composed, precisely, they put into effect another relation, conforming to the relation of arsenic: I have been poisoned! Good. Do you see that in one sense it is very serious, but it is not that serious, for Spinoza. Because, in the end, I can say that death ? concerns what? We can say in advance, before knowing what it is that he calls an essence, death concerns essentially a fundamental dimension of the individual, but a single dimension, that is the relationship of the parts to an essence. But it concerns neither the relation under which the parts belong to me, nor the essence. Why?
You‚ve seen that the characteristic relation, the differential relation, or the differential relations which characterise me, they are independent in themselves, they are independent of the terms since the terms are infinitely small, and that the relation, itself, on the contrary, has a finite value: dy/dx = z. Then it is actually true that my relation or my relations cease to be put into effect when I die, there are no longer parts which effect. Why? Because the parts have been set up to put into effect other relations. Good. But firstly there is an eternal truth of the relation, in other words there is a consistency of the relation even when it is not put into effect by actual parts, there is an actuality of the relation, even when it ceases to be put into effect. That which disappears with death is the effectuation of the relation, it is not the relation itself. You say to me: what is a non-effectuated relation? I call upon this logic of the relation such as it seems to be born in the 17th century, that is to say it has effectively shown in what conditions a relation had a consistency while its terms were vanishing. There is a truth of the relation independent of the terms which put the relation into effect, and on the other hand there is a reality of the essence that is expressed in the relation, there is a reality of the essence independent of knowing if the actually given parts putting the relation into effect conform with the essence. In other words both the relation and the essence are said to be eternal, or at least to have a species of eternity ˜ a species [espèce] of eternity doesn‚t at all mean a metaphoric eternity ˜ it is a very precise type of eternity, that is to say that: the species of eternity in Spinoza has always signified what is eternal by virtue of its cause and not by virtue of itself ˜ therefore the singular essence and the characteristic relations in which this essence expresses itself are eternal, while what is transitory, and what defines my existence, is uniquely the time during which the infinitely small extensive parts belong to me, that is to say put the relation into effect. But then there you are, this is why it is necessary to say that my essence exists when I don‚t exist, or when I no longer exist. In other words there is an existence of the essence which is not confused with the existence of the individual whose essence is the essence in question. There is an existence of the singular essence which is not confused with the existence of the individual whose essence is the essence in question. It is very important because you see where Spinoza is heading, and his whole system is founded on it: it is a system in which everything that is is real. Never, never has such a negation of the category of possibility been carried so far. Essences are not possibilities. There is nothing possible, everything that is is real. In other words essences don‚t define possibililties of existence, essences are themselves existences.
Here he goes much further than the others of the 17th century ˜ here I‚m thinking of Leibniz. With Leibniz, you have an idea according to which essences are logical possibilities. For example, there is an essence of Adam, there is an essence of Peter, there is an essence of Paul, and they are possibles. As long as Peter, Paul, etc. don‚t exist, we can only define the essence as a possible, as something which is possible. Simply, Leibniz will be forced, henceforth, to give an account of this: how can the possible account for, integrate in itself the possibility of existing, as if it would be necessary to burden the category of the possible with a kind [espèce] of tendency to existence. And, in effect, Leibniz develops a very very curious theory, with a word that is common to both Leibniz and Spinoza, the word conatus, tendency, but which actually has two absolutely different senses in Spinoza and Leibniz. With Leibniz singular essences are simply possibles, they are special possibles since they tend with all their force to exist. It is necessary to introduce into the logical category of possibility a tendency to existence.
Spinoza, I‚m not saying that it is better ˜ it‚s your choice ˜ it is truly a characteristic of the thought of Spinoza, for him, it is the same notion of the possible: he doesn‚t want to enrich the notion of the possible by grafting it to a tendency to existence. What he wants is the radical destruction of the category of the possible. There is only the real. In other words, essence isn‚t a logical possibility, essence is a physical reality. It is a physical reality, what can that mean? In other words, the essence of Paul, once Paul is dead, remains a physical reality. It is a real being. Therefore it would be necessary to distinguish them as two real beings: the being of the existence and the being of the essence of Paul. What‚s more, it would be necessary to distinguish as two existences: the existence of Paul and the existence of the essence of Paul. The existence of the essence of Paul is eternal, while the existence of Paul is transitory, mortal, etc. You see, from the point where we‚re at, if this is it, a very important theme of Spinoza is: but what is it going to be, this physical reality of the essence? Essences can‚t be logical possibilities, if they were logical possibilities they would be nothing: they must be physical realities. But be careful, these physical realities are not confused with the physical reality of the existence. What is the physical reality of the existence? Spinoza finds himself in the grip of a problem which is very very complicated, but so much the better. I want this all to be clear, I don‚t know how to do it.
Spinoza tells us, I‚ll tell you shortly when and where he tells us this, in a very fine text, he tells us: imagine a white wall. A totally white wall. There is nothing on it. Then you arrive with a pencil, you draw a man, and then next to it you do another. There your two men exist. They exist insofar as what? They exist insofar as you‚ve traced them. Two figures exist on the white wall. You can call these two figures Peter and Paul. So long as nothing is traced on the white wall, does something exist which would be distinct from the white wall? Spinoza‚s response very curiously is: ŒNo, strictly speaking nothing exists!‚ On the white wall, nothing exists so long as the figures haven‚t been traced. You‚re telling me that this isn‚t complicated. It isn‚t complicated. It is a fine example because I will have need of it for the next time. As for now, all I will do is comment on Spinoza‚s text. Now, where can this text be found? This text can be found in the early work of Spinoza, a work which was not written by him, but is the notes of an auditor, and is known by the title the Short Treatise. The Short Treatise. You see why this example is important. The white wall is something equivalent to what Spinoza calls the attribute. The attribute, extension. The question was asked: but what is there in extension? In extension there is extension, the white wall equals a white wall, extension equals extension! But you could say: bodies exist in extension. Yes, bodies exist in extension. OK. What is the existence of bodies in extension? The existence of bodies in extension is effectively when these bodies are traced. What does that mean, effectively traced? We have seen this answer, Spinoza‚s very strict answer, it is when an infinity of infinitely small parts are determined to belong to a body. The body is traced. It has a shape. What Spinoza will call mode of the attribute is such a shape. Therefore the bodies are in extension exactly like figures traced on the white wall, and I can distinguish one figure from another figure, by saying precisely: which parts belong to which shape, pay attention, such another part, it can have there common fringes, but what can this do? This means that there will be a common relation between the two bodies, yes, this is possible, but I would distinguish existing bodies. Outside of that, can I distinguish something? One finds that the text of the Short Treatise, of Spinoza‚s youth, seems to say: ultimately it is impossible to distinguish something outside of existing modes, outside of shapes. If you haven‚t traced the shape, you cannot distinguish something on the white wall. The white wall is uniformly white. Excuse me for dwelling on this, it is really because it is an essential moment in Spinoza‚s thought. Nevertheless, already in the Short Treatise he says: the essences are singular, that is to say there is an essence of Peter and of Paul which is not confused with the existence of Peter and Paul. Now, if essences are singular, it is necessary to distinguish something on the white wall without the shapes necessarily having been traced. What‚s more, if I leap to his definitive work, the Ethics, I see that in Part II, Proposition 7, 8, etc., Spinoza comes across this problem again. He says, very bizarrely: modes exist in the attribute in two ways; on the one hand they exist insofar as they are comprised and contained in the attribute; and, on the other, insofar as it is said that they have duration. Two existences: durational existence, immanent existence. Here I take the letter of the text. Modes exist in two ways, that is to say that: existing modes exist insofar as they are said to have duration, and the essences of modes exist insofar as they are contained in the attribute. Good! This is complicated because modal essences are once again, and here it is confirmed by all the texts of the Ethics, they are singular essences, meaning that one isn‚t confused with the essence of another, the one isn‚t confused with the other, good, very well. But then, how are they distinguished in the attribute, one from the other. Spinoza affirms that they are distinguished, and then here he abandons us! Does he really abandon us, it is not possible! A thing like this is not imaginable! He doesn‚t tells us, OK. He gives an example, he gives us a geometric example, precisely, which comes down to saying: does a shape have a certain mode of existence when it isn‚t traced? Does a shape exist in extension when it isn‚t traced in extension? The whole text seems to say: yes, and the whole text seems to say: complete this yourselves. And this is normal, perhaps he has given us all the elements of an answer. To complete it ourselves. It is necessary, we have no choice! Either we renounce being Spinozist ˜ that wouldn‚t be bad either ˜ or it is necessary to complete it yourself. How could we complete this ourselves? This is why I pleaded as I said at the beginning of the year, we plead with ourselves, on the one hand, with the heart, and, on the other, with that which we know. The white wall! Why does he speak of the white wall? What is this story of the white wall?
After all, examples in philosophy are also a bit like a wink of the eye. You tell me: well what do we do if we don‚t understand the wink of an eye? It‚s not serious, not serious at all! We pass by a million things. Let‚s make do with what we have, let‚s make do with what we know. White wall. But after all I‚m trying to complete it with all my heart before completing it with knowledge. Let‚s call on our hearts. I take on one side my white wall, on the other side my drawings on the white wall. I have drawn on the wall. And my question is this: can I distinguish on the white wall things independently of the shapes drawn, can I make distinctions which are not distinctions between shapes?
Here it is like a practical exercise, there is no need to know anything. Simply, I say: you are reading Spinoza well if you arrive at this problem or at an equivalent problem. It is necessary to read him literally enough in order for you to say: ah yes, this is the problem that he poses us, and the job for him is to pose the problem so precisely that ˜ it is even a present that he gives us in his infinite generosity ˜ is to pose the problem so well, he poses the problem for us so precisely that obviously, we tell ourselves, the answer is this, and we will have the impression of having found the answer. It is only the great authors who give you this impression. They stop just when all is finished, but no, there is a tiny bit that they haven‚t mentioned. We are forced to find it and we say to ourselves: I am good aren‚t I, I am strong aren‚t I, I found it, because at the moment when I come to pose the question like this: can something be distinguished on the white wall independently of the drawn shapes? It is obvious that I have the answer already. And that we respond in chorus, we respond: Of course, there is another mode of distinction which is what? It is that the white has degrees! And I can vary the degrees of whiteness. One degree of whiteness is distinguished from another degree of whiteness in a totally different way than that by which a shape on the white wall is distinguished from another shape on the white wall. In other words, the white has, one says in Latin ˜ we use all the languages in order to try to better understand the languages that we don‚t know, what! (laughs) ˜ the white has distinctions of gradus. There are degrees, and the degrees are not confused with the shapes. You say: such a degree of white, in the sense of such a degree of light. A degree of light, a degree of whiteness, is not a shape. And even though two degrees are distinguished, two degrees aren‚t distinguished like shapes in space. I would say that shapes are distinguished externally, taking account of their common parts. I would say of degrees that it is a completely different type of distinction, that there is an intrinsic distinction. What is this?
Accordingly I no longer need ? It is an accident. Each operates with what they know. I tell myself: ha, it is not at all surprising that Spinoza ? What is the wink of the eye from the point of view of knowledge? We started with our heart by saying: yes, it can only be that: there is a distinction of degrees which is not confused with the distinction of figures. The light has degrees, and the distinction of degrees of light is not confused with the distinction of shapes in the light. You tell me that all of this is infantile; but it is not infantile when we try to make them philosophical concepts. Yes it‚s infantile, and it isn‚t. That‚s good. Well then, what is this story, there are intrinsic distinctions! Good, let‚s try to progress, from the point of view of terminology. It is necessary to make a terminological grouping.
My white wall, the white of the white wall, I will call: Œquality‚. The determination of shapes on the white wall I will call: Œmagnitude, or length‚. ˜ I will say why I use the apparently bizarre word magnitude [grandeur]. Magnitude, or length, or extensive quantity. Extensive quantity is in effect the quantity which is composed of parts. Recall the existing mode, existing me, is defined precisely by the infinity of parts which belong to me. What else is there besides quality, the white, and extensive quantity, magnitude or length, there are degrees. There are degrees which are what, which we call in general: intensive quantities, and which are in fact just as different from quality as from extensive quantity. These are degrees or intensities.
Now there is a philosopher of the Middle Ages, one of great genius, here I appeal to a very small bit of knowledge, he is called Duns Scotus. He appeals to the white wall. It is the same example. Did Spinoza read Duns Scotus? [This is] of no interest, because I am not sure at all that it is Duns Scotus who invented this example! It is an example which can be found throughout the Middle Ages, in a whole group of theories of the Middle Ages. The white wall. Yes! ... He said: quality, the white, has an infinity of intrinsic modes. He wrote in Latin: modus intrinsecus. And Duns Scotus here innovated, invented a theory of intrinsic modes. A quality has an infinity of intrinsic modes. Intrinsic modes, what are they, and he says: the white has an infinity of intrinsic modes, these are the intensities of white. Understand: white equals light in the example. An infinity of luminous intensities. He adds this ˜ and note that he takes responsibility because here this is new ˜ you tell me, say: there is an intensity, there is an infinity of intensities of light. Ok, not much. But what does he draw from this and why does he say this? What accounts does he settle, and with whom? This becomes important. Understand that the example is typical because when he says white, or quality, he means as well: Œform‚. In other words, we are in open discussion of the philosophy of Aristotle, and he is saying: a form has intrinsic modes. Ha! If he means: a form has intrinsic modes, it doesn‚t go without saying, not at all! Why? Because, it goes without saying that all sorts of authors, all sorts of theologians would consider that a form would be invariable in itself, and that only existing things vary in which form puts itself into effect. Duns Scotus tells us: here where the others distinguish two terms, it is necessary to distinguish three: that in which the form puts itself into effect are extrinsic modes. Therefore it is necessary to distinguish form from extrinsic modes, but there is something else. A form has also a kind [espèce] ˜ as they say in the Middle Ages ˜ a kind of latitude, a latitude of form, which has degrees, the intrinsic degrees of form. Good. These are intensities, therefore intensive quantities. What distinguishes them? How is one degree distinguished from another degree? Here, I insist on this because the theory of intensive quantities is like the conception of differential calculus of which I have spoken, it is determinant throughout the whole of the Middle Ages. What‚s more, it is related to problems of theology, there is a whole theory of intensities at the level of theology. If there is a unity of physics and metaphysics in the Middle Ages, it is very centred ˜ understand this makes the metaphysics of the Middle Ages much more interesting, there is a whole problem of the Trinity, that is to say three persons for one and the same substance, that which obstructs the mystery of the trinity. It is always said: they fight like that, they are theological questions. Not at all, they are not theological questions, it engages everything because, at the same time: they determine a physics of intensities, in the Middle Ages; they determine an elucidation of theological mysteries, the Holy Trinity; and they determine a metaphysics of forms, all of this is way beyond the specificity of theology. Under what form are the three persons of the Holy Trinity distinguished? It is obvious that there is a kind of problem of individuation that is very very important. It is necessary that the three persons are, in a way, not at all different substances, it is necessary that they are intrinsic modes. Therefore how will they be distinguished? Have we not forged ahead here into a kind of theology of intensity.
When Klossowski in his literature finds a kind of very very strange connection between theological themes of which it is said: but after all where does all of this come from, and a very Nietzschean conception of intensities, it would be necessary to see, as Klossowski is someone extremely wise, erudite, it is necessary to see what link he makes between these problems of the Middle Ages and current questions or the Nietzschean questions. It is obvious that in the Middle Ages the whole theory of intensities is simultaneously physical, theological and metaphysical. Under which form?
[end of tape ˜ very little time before the end of the course.]